Optimal. Leaf size=583 \[ -\frac{4 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b} \]
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Rubi [A] time = 0.733102, antiderivative size = 583, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1888, 1594, 1886, 261, 1878, 218, 1877} \[ -\frac{4 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b} \]
Antiderivative was successfully verified.
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Rule 1888
Rule 1594
Rule 1886
Rule 261
Rule 1878
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x+e x^2\right )}{\sqrt{a+b x^3}} \, dx &=\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{2 \int \frac{-3 a e x^2+\frac{9}{2} b c x^3+\frac{9}{2} b d x^4}{\sqrt{a+b x^3}} \, dx}{9 b}\\ &=\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{2 \int \frac{x^2 \left (-3 a e+\frac{9 b c x}{2}+\frac{9}{2} b d x^2\right )}{\sqrt{a+b x^3}} \, dx}{9 b}\\ &=\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{4 \int \frac{-9 a b d x-\frac{21}{2} a b e x^2+\frac{63}{4} b^2 c x^3}{\sqrt{a+b x^3}} \, dx}{63 b^2}\\ &=\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{4 \int \frac{x \left (-9 a b d-\frac{21}{2} a b e x+\frac{63}{4} b^2 c x^2\right )}{\sqrt{a+b x^3}} \, dx}{63 b^2}\\ &=\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{8 \int \frac{-\frac{63}{4} a b^2 c-\frac{45}{2} a b^2 d x-\frac{105}{4} a b^2 e x^2}{\sqrt{a+b x^3}} \, dx}{315 b^3}\\ &=\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}+\frac{8 \int \frac{-\frac{63}{4} a b^2 c-\frac{45}{2} a b^2 d x}{\sqrt{a+b x^3}} \, dx}{315 b^3}-\frac{(2 a e) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{3 b}\\ &=-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}-\frac{(4 a d) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{7 b^{4/3}}-\frac{\left (2 a \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{35 b^{4/3}}\\ &=-\frac{4 a e \sqrt{a+b x^3}}{9 b^2}+\frac{2 c x \sqrt{a+b x^3}}{5 b}+\frac{2 d x^2 \sqrt{a+b x^3}}{7 b}+\frac{2 e x^3 \sqrt{a+b x^3}}{9 b}-\frac{8 a d \sqrt{a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2+\sqrt{3}} a \left (7 \sqrt [3]{b} c-10 \left (1-\sqrt{3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.128637, size = 132, normalized size = 0.23 \[ \frac{-2 \left (a+b x^3\right ) (70 a e-b x (63 c+5 x (9 d+7 e x)))-126 a b c x \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a}\right )-90 a b d x^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{315 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 793, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d x + c\right )} x^{3}}{\sqrt{b x^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e x^{5} + d x^{4} + c x^{3}}{\sqrt{b x^{3} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.13196, size = 129, normalized size = 0.22 \begin{align*} e \left (\begin{cases} - \frac{4 a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases}\right ) + \frac{c x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{7}{3}\right )} + \frac{d x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{8}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d x + c\right )} x^{3}}{\sqrt{b x^{3} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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